Suppose now that we are in case (II). If two geodesic segments issue from the same point and have angle \(\pi \)  between them, then by [1]} the concatenation of the two segments is also a geodesic. It follows that the geodesic \([p_i(x),p_j(x)]\)  is the concatenation of the geodesics \([p_i(x),b_{ij}]\)  and \([b_{ij},p_j(x)]\) , where \(b_{ij}\)  is a fixed choice of either endpoint of \(B_{ij}\) . Consider the geodesic quadrilateral defined by \(x, p_i(x),b_{ij}\)  and \(p_j(x)\) . The angle at \(b_{ij}\)  is exactly \(\pi \)  and, as before, the angles at \(p_i(x)\)  and \(p_j(x)\)  are at least \(\frac{\pi }{2}\) . It follows from the Flat Quadrilateral Theorem [1]} that the angle at \(x\)  is zero and therefore (after possibly swapping \(i\)  and \(j\) ) we have that \(p_i(x)\)  lies on the geodesic \([x,p_j(x)]\) . Again, this gives \(p_j(x)=p_j(p_i(x))\in D_j \subseteq A_j \backslash A_j^+\) , which is a contradiction.
